Global Boundedness Of Chemotaxis Model With Logistic Growth And Indirect Signal

Chemotaxis is a common phenomenon in ecology and biology,which describes the directed motion of biological population cells under the influence of some chemical signal substances.Chemotaxis plays an important role in the survival and development of cells.Since Keller and Segel proposed the famous Keller-Segel model in 1970,more and more scholars begin to use mathematical methods to analyze and explain this phenomenon.In this paper,we mainly consider the global boundedness and long-time behavior of the solution of the following full-parabolic chemotaxis system with logistic growth and indirect signal production(?)under homogeneous Neumann boundary conditions,where Ω(?)R~d(d≥2)is a bounded domain with smooth boundary,the parameters μ>0,α>1,n(x,t)denotes the cell density,c(x,t),w(x,t)and z(x,t)denote the concentrations of three chemicals,respectively.Our main results are as follows:if α>max {1,d/6+1/3},then for sufficiently smooth initial data,the above model admits a unique uniformly bounded global classical solution.Moreover,if μ>0 is large enough,the solution converges to a constant equilibrium.For the proof,we will first obtain the local existence of the solution by using the standard fixed point method,then establish a priori estimates of n,c,w,z and extend the local solution to a global one by the extension criterion.Finally,we establish the long-time behavior of the solution by using the modified energy functional.